where \(N_k\)  is the no. of efolds since horizon exit to the end of inflation. Typically, we have \(H_{end} \simeq 2 \times 10^{-9} m_P\) , while for example \(H_k \simeq 2 \times 10^{-8}\)  for \(N_k \simeq 5\) , so the PBHs will have masses in the range \(5 \times 10^4 ~{\rm g} \lesssim M_{PBH} \lesssim 10^6 ~{\rm g}\) , and will therefore evaporate before BBN [1]}, [2]}, [3]}, [4]}. Still, they can produce relics that might overclose the universe, and to avoid this (model dependent) possibility, one may impose an upper limit on their mass fraction \(\beta (M) \lesssim 10^{-16}\) . Using the Press-Schechter formalism, for a Gaussian distribution of fluctuations \(\delta =\delta \rho / \rho \)  the mass fraction is given by:
\(\beta (M) = \int _{\delta _c}^\infty d\delta P(\delta ) = \frac{2}{\sqrt{2 \pi } \sigma (M)} \int _{\delta _c}^\infty d\delta {\rm exp}( -\frac{\delta ^2}{\sigma (M)^2}) = {\rm erfc}(\frac{\delta _c}{\sqrt{2}\sigma (M)}) \,, \) 
